Visualizing Nodal Heavy Fermion Superconductivity

(From Left to Right) Brian Zhou, Shashank Misra, and Ali Yazdani

Authors: Brian Zhou, Shashank Misra, and Ali Yazdani

Affiliation: Joseph Henry Laboratories and Department of Physics, Princeton University, USA

Link to Yazdani Lab >>

Superconductors, broadly speaking, can be divided into two classes by how its electrons bind together to form the “Cooper pairs” that sustain dissipation-less current flow. The first class of “conventional” superconductors contains all of the metallic superconductors, such as mercury, lead, or niobium. Here the attractive potential, or pairing symmetry (represented by a circle in Fig. 1a), of the Cooper pair is equally strong for all electrons, and transition temperature T_c at which superconductivity emerges is limited to at most 30 K, only one tenth of room temperature. The second class of “unconventional”, or extraordinary, superconductors behave much differently – and strikingly can possess transition temperatures in excess of 150 K, half of room temperature. The pairing potential (represented by a four-leaf clover in Fig. 1b) now depends sensitively on the direction of the electron’s momentum, with certain directions (called nodes) having zero pairing strength.

Fig.1 a) Representation of the pairing symmetry of a conventional superconductor by a circle. The paring amplitude is isotropic in direction. b) Representation of the pairing symmetry for an unconventional (d_x²-y²) superconductor by a clover shape. The pairing amplitude is zero for certain directions called the nodes, depicted by the dashed lines.

Heavy fermion materials, whose electrons behave as if they have greatly enhanced mass due to strong interactions between them, belong to this unconventional class of superconductors that also includes the widely known high-T_c copper-oxide “cuprate” superconductors. Intense research effort has determined the pairing symmetry of the cuprate superconductors to be of d_x²-y² form (clover leaves separated by nodes occurring diagonal to the axis); however, an understanding of its mechanism and the precise cause of their high transition temperatures remain elusive [1]. Heavy fermions, such as CeCoIn₅, have been suggested to share the same paring symmetry, and thus possibly the same underlying mechanism, as the cuprate superconductors [2]. Nevertheless, the superconducting phase of heavy fermions has largely dodged the same intense experimental spotlight, as their low transition temperatures preclude access by conventional experimental tools such as photoemission (ARPES) and, until recently, scanning tunneling microscopy (STM) [3, 4].

To probe superconductivity in the prototypical heavy fermion CeCoIn₅, synthesized by our collaborators at Los Alamos National Laboratory, we developed a new state-of-the-art STM operating at dilution-fridge temperature and high magnetic field. Like the high-T_c cuprates, superconductivity in CeCoIn₅ emerges in close proximity to an anti-ferromagnetic critical point and out of a normal state showing strong non-Fermi liquid behavior [5]. Accordingly, angle-resolved thermal transport and neutron scattering experiments [6, 7] have shown data consistent with a d_x²-y² gap symmetry; however, until our experiment, no phase sensitive identification has been performed. Unlike the high-T_c cuprates and more favorable to experiment, unconventional superconductivity occurs in the undoped, and thus ultra-clean, sample of CeCoIn₅ and can be extinguished here by an experimentally-accessible magnetic field (5 T perpendicular to the c-axis).

STM can probe the pairing symmetry of an unconventional superconductor through two ways: 1) studying the surface electron waves caused by the interference of quasiparticles scattering off impurities, or 2) studying the bound state formed by a pair-breaking impurity. Our research is the first to perform this comprehensive set of experiments on superconducting CeCoIn₅, reaching a temperature of 245 mK, well below CeCoIn₅’s Tc of 2.3 K. To isolate the salient effects of superconductivity, we in fact conducted each experiment first at zero magnetic field, where superconductivity is strongest, and then repeated each experiment again at 5.7 T, where superconductivity is extinguished. The quasi-particle interference (QPI) patterns reveal heavy bands (m* ~ 25 m₀) in the normal state that rapidly close within ~10 mV above the Fermi level (see Fig. 2). These heavy bands develop significant changes with the onset of superconductivity; however, the particular fashion in which these changes occur in our data cannot be explained by the simplest (electron-hole symmetric) model of QPI that has previously been applied to the cuprate superconductors. This is unsurprising because the Fermi surface of CeCoIn₅ possesses multiple bands with complicated shapes, in contrast to the typical single circular hole barrel of the cuprate Fermi surface. Consequently, conclusions drawn from the superconducting QPI would require a large number of assumptions that cannot be independently justified.

Fig. 2 a) Electron waves formed by the interference of heavy electrons on the surface of CeCoIn₅. b) The discrete Fourier transform of the real space conductance map in a) gives information on the wavelength and direction of the waves seen in the image. By performing these measurements as a function of energy, we determine the relation between energy and momentum of the heavy electrons that pair together to produce superconductivity, showing them to behave with effective masses up to 30 times the bare electron mass.

While the QPI in CeCoIn₅ could not provide an unambiguous result, the spatial distribution of an impurity bound state fortuitously revealed a direct nanoscale fingerprint of the superconducting gap symmetry. Excitations in a superconductor are so called Boguliubov quasiparticles, which are mixtures of electrons and holes in the Fermi sea. The spectrum in Fig 3a, with enhancement at positive bias, shows an impurity attractive to the electron part of the Boguliubov quasiparticle. This electron-like component leaks out away from the impurity along the minima of pairing potential, thereby pinpointing the nodes to occur at diagonal to the crystal axis in d_x²-y² form [8,9]. We can further confirm the identity of this signal by imaging the hole component of the same impurity state, showing it to be spatially complementary to the electron component as is expected for superconducting pairing. As a final check of its superconducting origin, this bound state disappears when superconductivity is extinguished.

Fig. 3 An impurity in an unconventional superconductor locally disturbs the Cooper pairs of the electron sea. In a), we determine that this particular impurity attracts the electron component of the Cooper pair as its spectrum shows an enhancement at positive energies relative to the spectrum far away from it. The characteristic energy-dependent spatial patterns of how this impurity locally perturbs superconductivity (shown for negative and positive energy in b) and c), respectively) directly reveal the symmetry of unconventional Cooper pairing in this compound.

Our research on CeCoIn₅ firmly establish its pairing symmetry to parallel that of the high-T_c cuprate superconductors and extend for the first time the power of the STM to another class of extraordinary superconductors, the heavy fermions. In fact, the parallel to the high-T_c goes even further: our spectroscopy reveals superconductivity to develop within a depression of the density of states near the Fermi level that persists above T_c and above the critical magnetic field. Could this gap have a similar origin as the peculiar normal state “pseudogap” in the cuprates? Most importantly, the ability to study CeCoIn₅ broadens the experimental tool-kit for tackling the questions of unconventional superconductivity – what role of magnetism, other competing phases, and electron-electron interaction in the normal state play in making these special superconductors the most robust of the bunch.

References:
[1] C.C Tsuei, J.R. Kirtley, "Pairing symmetry in cuprate superconductors". Review of Modern Physics, 72, 969-1016 (2000). Abstract.
[2] Joe D. Thompson and Zachary Fisk. "Progress in Heavy-Fermion Superconductivity: Ce115 and Related Materials". Journal of the Physical Society of Japan, 81, 011002 (2012). Abstract.
[3] Brian B. Zhou, Shashank Misra, Eduardo H. da Silva Neto, Pegor Aynajian, Ryan E. Baumbach, J. D. Thompson, Eric D. Bauer, Ali Yazdani. "Visualizing nodal heavy fermion superconductivity in CeCoIn₅". Nature Physics, 9, 474-479 (2013). Abstract.
[4] M. P. Allan, F. Massee, D. K. Morr, J. Van Dyke, A. W. Rost, A. P. Mackenzie, C. Petrovic, J. C. Davis. "Imaging Cooper pairing of heavy fermions in CeCoIn₅". Nature Physics, 9, 468-473 (2013). Abstract.
[5] Pegor Aynajian, Eduardo H. da Silva Neto, András Gyenis, Ryan E. Baumbach, J. D. Thompson, Zachary Fisk, Eric D. Bauer & Ali Yazdani. "Visualizing heavy fermions emerging in a quantum critical Kondo lattice". Nature, 486, 201-206 (2012). Abstract,
[6] K. Izawa, H. Yamaguchi, Yuji Matsuda, H. Shishido, R. Settai, and Y. Onuki. "Angular position of nodes in the superconducting gap of quasi-2D heavy-fermion superconductor CeCoIn₅". Physical Review Letters, 87, 057002 (2001). Abstract.
[7] C. Stock, C. Broholm, J. Hudis, H. J. Kang, and C. Petrovic. "Spin Resonance in the d-Wave Superconductor CeCoIn₅". Physical Review Letters, 100, 087001 (2008). Abstract.
[8] Stephan Haas, Kazumi Maki. "Quasiparticle bound states around impurities in d_x²-y²-wave superconductors". Physical Review Letters, 85, 2172-2175 (2000). Abstract.
[9] A.V. Balatsky, I. Vekhter, J.X. Zhu. "Impurity-induced states in conventional and unconventional superconductors". Review of Modern Physics, 78, 373-433 (2006). Abstract.

Deterministic Quantum Teleportation

Christine A. Muschik (left) 
and 
Eugene S. Polzik (right)













Authors: Christine A. Muschik¹ and Eugene S. Polzik²

Affiliation:
¹ICFO - Institut de Ciències Fotòniques, Barcelona, Spain.
²Niels Bohr Institute and Danish Quantum Optics Center QUANTOP, Copenhagen University, Denmark.

Teleportation [1] is an essential element in quantum information science [2] and is intimately linked to and enabled by the distribution of entanglement between two locations.

The transmission of quantum information is a tricky business, since quantum systems are typically too fragile to be sent directly over large distances. Moreover, the no-cloning theorem [3] prevents one from realizing a faithful transmission by measuring a quantum state in one location and re-preparing it in another place.

Since teleportation provides a possibility to circumvent these problems, it is a key-ingredient in distributed quantum networks and lies at the heart of the practical realization of long distance quantum communication [4]. Apart from that, teleportation allows for universal quantum computing [5] if supplemented with local operations and suitable entangled resource states.

The first experimental realizations of teleportation protocols employed light as the carrier of quantum states [6]. The teleportation between matter systems is more challenging, but also particularly relevant. In contrast to photons, matter systems provide long-lived degrees of freedom which are required for the storage of quantum information. Matter systems are also needed for the efficient processing of quantum states and the extension to large quantum networks which involve multiple remote nodes.

The teleportation of quantum systems between matter systems has been demonstrated deterministically between ions that are held in the same trap [7]. However, for macroscopic distances, so far only probabilistic implementations were available, which yield random outcomes and succeed therefore only with a certain probability.

In our recent publication [8], we report on the realization of a novel protocol for teleportation over a macroscopic distance [9], which allows for the deterministic transfer of a quantum state, i.e. this scheme guarantees the successful teleportation of quantum states in every single attempt. This feature is important for technological applications and also opens up new possibilities for the teleportation of quantum dynamics [10].

The experiment is carried out using two atomic clouds, A and B, at room temperature which are contained in glass cells that are placed 0.5 meter apart [11]. Each atomic cloud contains approximately 10¹² Cesium atoms and the quantum state of each of these samples is encoded in the collective atomic spin state of the atomic cloud. A freely propagating laser field is used to entangle the two atomic samples and to teleport the spin state of cloud B to cloud A.

Figure 1: Deterministic quantum teleportation between two atomic clouds. A laser field is used to entangle the two samples and to teleport the spin state of cloud B to cloud A.

More specifically, the interaction between the laser field and a gas sample leads to entanglement between the atoms and the light. In our teleportation protocol, the laser field passes both samples, A and B, and is afterwards measured. This procedure first creates entanglement between the sample A and the light. The light then passes through the sample B and gets an imprint of its quantum state. The teleportation scheme is completed by performing a conditional feedback operation on cloud A: the spin state of sample A is displaced according to the measurement outcome using magnetic fields that are applied to the container of cloud A.

We used the deterministic character of the teleportation scheme to realize a “stroboscopic” teleportation, in which a sequence of rapidly changing spins states has been teleported between the two gas samples. This approach can be further extended to achieve a truly time-continuous teleportation, which paves the way toward the teleportation of quantum dynamics and the simulation of interactions between distant objects which cannot interact directly [10].

This work was done in collaboration with H. Krauter, D. Salart, J. M. Petersen, and H. Shen at the Niels Bohr Institute, Copenhagen, Denmark and T. Fernholz at the University of Nottingham, UK.

References:
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels", Physical Review Letters, 70, 1895 (1993). Abstract.
[2] Michael A. Nielsen, Isaac L. Chuang, "Quantum Computation and Quantum Information". Cambridge University Press, Cambridge (2000).
[3] W.K. Wootters and W.H. Zurek, "A single quantum cannot be cloned", Nature, 299, 802 (1982). Abstract.
[4] H. J. Kimble, "The Quantum Internet", Nature, 453, 1023 (2008). Abstract.
[5] Daniel Gottesman and Isaac L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations", Nature, 402, 390 (1999). Abstract.
[6] Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter & Anton Zeilinger, "Experimental quantum teleportation", Nature, 390, 575 (1997). Abstract; A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, "Unconditional Quantum Teleportation", Science 282, 706 (1998). Abstract.
[7] M. Riebe, H. Häffner, C.F. Roos, W. Hänsel, J. Benhelm, G.P.T. Lancaster, T.W. Körber, C. Becher, F. Schmidt-Kaler, D.F.V. James, R. Blatt, "Deterministic quantum teleportation with atoms", Nature 429, 734 (2004). Abstract  M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, D. J. Wineland, "Deterministic quantum teleportation of atomic qubits", Nature 429, 737 (2004). Abstract ; M Riebe, M Chwalla, J Benhelm, H Häffner, W Hänsel, C F Roos and R Blatt, "Quantum teleportation with atoms: quantum process tomography", New Journal of Physics, 9, 211 (2007). Abstract.
[8] H. Krauter, D. Salart, C. A. Muschik, J. M. Petersen, Heng Shen, T. Fernholz, and E. S. Polzik, "Deterministic quantum teleportation between distant atomic objects", Nature Physics, 9, 400 (2013). Abstract.
[9] Christine A. Muschik, "Quantum information processing with atoms and photons". Ph.D. thesis, Max-Planck Institute for Quantumoptics (2011). PDF file.
[10] Christine A. Muschik, Klemens Hammerer, Eugene S. Polzik, and Ignacio J. Cirac, "Quantum Teleportation of Dynamics and Effective Interactions between Remote Systems", Physical Review Letters, 111, 020501 (2013). Abstract.
[11] Klemens Hammerer, Anders S. SØrensen, and Eugene S. Polzik, "Quantum interface between light and atomic ensembles", Review of Modern Physics, 82, 1041 (2010). Abstract.

Laser Cooling to Quantum Degeneracy

The SrBEC team in October 2012. From left to right: Slava M. Tzanova^1,2, Benjamin Pasquiou¹, Rudolf Grimm^1,2, Simon Stellmer^1,2,3 (author), Florian Vogl^1,2, Florian Schreck¹ (author), Alex Bayerle^1,2

Authors: Simon Stellmer^1,2,3 and Florian Schreck¹

Affiliation:
¹Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 6020 Innsbruck, Austria
²Institute for Experimental Physics and Center for Quantum Physics, University of Innsbruck, 6020 Innsbruck, Austria
³Institute for Atomic and Subatomic Physics, Vienna University of Technology, 1020 Vienna, Austria

Link to the Sr BEC project homepage >>           
Link to the “Ultracold Atoms and Quantum Gases” group in Innsbruck >>

Laser cooling is a very elegant and versatile technique, as it permits to cool atoms, molecules, ions, and even mechanical objects from room temperature down to temperatures as low as one millionth of a degree above absolute zero [1,2]. At these low temperatures, we can enter into the world of quantum mechanics. One of the fascinating phenomena associated with this ultracold regime is the appearance of quantum degeneracy in atomic gases [2]. Since the early days of laser cooling, the question has been asked if the quantum degenerate regime could be reached using laser cooling as the only cooling process. Despite significant experimental and theoretical effort to overcome the limitations of laser cooling this goal has been elusive.

Past 2Physics article by Florian Schreck:
November 29, 2009: "Bose-Einstein Condensation of Strontium"

In 1995, the combination of laser cooling with a subsequence stage of evaporative cooling led to the attainment of quantum degeneracy in bosonic alkali-metal gases [2,3]. These quantum gases, called “Bose-Einstein condensates” (BECs), have opened an extraordinary window for the exploration of the quantum world.

To create a BEC, the phase-space density of the gas has to be increased beyond a critical value by lowering the temperature and increasing the density of the gas. Laser cooling was so far unable to reach quantum degeneracy because the photons used to cool the gas have negative side effects, which limit the achievable density and destroy a BEC.

In our experiment we overcome these side effects and create a BEC of strontium by laser cooling [4]. Furthermore, our method creates a BEC immersed in a laser cooled cloud of atoms, which opens a simple path towards the construction of a truly continuous atom laser.

Our scheme relies on the combination of three techniques, favored by the properties of strontium. The first technique is called “narrow line cooling”. Strontium possesses a narrow laser cooling transition of only 7 kHz width. Operating a magneto-optical trap (MOT) on this narrow line, we can cool a gas of strontium atoms down to temperatures of about 800 nK. This is already more than an order of magnitude colder than conventional MOTs of alkali atoms!

The second technique is a separation of our cold gas into two spatial regions (see Fig. 1): one large region, in which about 10 million atoms are trapped in a “reservoir” optical dipole trap and continuously cooled by laser light, and a small region, in which about 1 million atoms are confined at a much higher density by a steep confining potential, often called a “dimple”. This is the region where the BEC will be created.

Figure 1: Three absorption images of 10 million strontium atoms trapped in a dipole trap and cooled to about 800nK by laser light. All images are taken on the narrow cooling transition. The left image shows the reservoir of atoms, which is used to dissipate heat. For the central image, we have applied the transparency beam, such that atoms located within this beam cannot absorb photons from the cooling light. The density in this region is greatly enhanced by a dimple beam, as can be seen on the right image, where the transparency beam has been turned off.

The third technique allows us to overcome the negative side effects of laser cooling photons in the dimple region. We protect atoms in this region from those photons by the help of an extra laser beam, which we call the “transparency beam”. This beam acts like a cap of invisibility, as it modifies the energy states of the atoms in the dimple region such that they cannot absorb laser cooling photons (see Fig. 1). Importantly, the atoms are not only transparent to the cooling laser beam, but also to laser cooling photons scattered towards the dimple region from atoms in the laser cooled reservoir.

Now we have two different regions: the outside “reservoir” region, in which atoms are gently cooled by laser light and a central dimple region, in which the BEC will form. A connection between the two regions is established through the elastic scattering between atoms: in this way, heat can be transferred very rapidly (on timescales of a few 10 ms) from the dimple into the reservoir, where the heat is dissipated. To maximize this heat transfer, we place the dimple right into the center of the reservoir, as can be seen in Fig. 1.

Once the system is prepared in this configuration, it takes only about 60 ms for a BEC to appear, and after a little over 100 ms, the BEC has reached its final size of about 100 000 atoms (see Fig. 2).

Figure 2: Absorption images, taken 24 ms after release from the trap. On the left image, the BEC is faintly visible as an elliptic density increase in the center. For the right image, we have removed all atoms from the reservoir just before the release from the trap, and the BEC stands out clearly.

An important property of our system is that laser cooling constantly provides strong dissipation, removing entropy from the gas. Even if we destroy our BEC by local heating of the dimple region, it will quickly reform after the heating process is switched off, as long as enough atoms are contained in the dipole trap.

We believe that our scheme can be adapted to other elements. We expect it to work with all species that possess a narrow cooling transition and have a reasonable scattering behavior. These criteria are fulfilled by a selection of elements, most prominently the lanthanides. The range of suitable candidates can be increased substantially by going one step further: sympathetic cooling between strontium and another element. This element would not need to feature a narrow cooling transition. Instead, it would be trapped in the dimple region and sympathetically cooled through collisions with the strontium atoms in the reservoir. We have recently implemented this sympathetic laser cooling scheme in a mixture of rubidium and strontium. The rubidium gas is cooled very efficiently by thermal contact with laser cooled strontium atoms, delivering ideal starting conditions for the creation of quantum degenerate Rb-Sr mixtures by evaporative cooling [5]. By using a Rb specific dipole trap as a dimple, it should also be possible to create a Rb BEC without evaporative cooling.

Our scheme also paves a relatively simple path towards a truly continuous atom laser. Here, a thermal source of atoms would be converted into a coherent beam of atoms, constantly outcoupled from the dimple region. The dimple would be continuously fed by the reservoir region, which in turn would be replenished by pre-cooled atoms from a MOT. Such truly continuous atom lasers are highly desired in various schemes of precision measurements.

References :
[1] Proceedings of the International School of Physics "Enrico Fermi", Course CXVIII, Varenna, 9-19 July 1991, Laser Manipulation of Atoms and Ions, edited by E. Arimondo, W. D. Phillips, and F. Strumia (North Holland, Amsterdam, 1992).
[2] Physics 2000, BEC homepage. Link.
[3] Proceedings of the International School of Physics ‘‘Enrico Fermi’’, Course CXI, Varenna, 7-17 July 1998, Bose-Einstein Condensation in Atomic Gases, edited by M. Inguscio, S. Stringari, and C. E. Wieman (North Holland, Amsterdam, 1999).
[4] Simon Stellmer, Benjamin Pasquiou, Rudolf Grimm, and Florian Schreck, "Laser Cooling to Quantum Degeneracy", Physical Review Letters, 110, 263003 (2013). Abstract.
[5] Benjamin Pasquiou, Alex Bayerle, Slava M. Tzanova, Simon Stellmer, Jacek Szczepkowski, Mark Parigger, Rudolf Grimm, and Florian Schreck, "Quantum degenerate mixtures of strontium and rubidium atoms", Physical Review A, 88, 023601 (2013). Abstract.